Mr Wilson, On Convection of Heat. 



407 



going into the parallelipiped in unit time through the face AyAz 

 in the direction of the axis x is 



AyAz 



^-^-)-*(™- 



where denotes the temperature at P. The heat going out 

 through the opposite face is 



AyAz 



■i^*£H-*S + iS*- 



The first -term in the brackets represents the convection of heat 

 and the second the conduction. We suppose that is measured 

 in degrees centigrade, and that the amount of heat in the medium 

 per unit volume is sp0. This amounts to taking the amount of 

 heat in unit volume as zero at 0° C. Since we are only concerned 

 with variations in the amount of heat present in unit volume this 

 latter assumption is legitimate. 



The difference between the above two expressions gives the 

 heat gained by the parallelipiped and is 



Ax AyAz 



dx 2 dx 



Similar expressions for the other two pairs of faces may be 

 obtained and the sum of all three is 



Ax AyAz 



\dx 2 dy 2 dz 2 



fd (p6u) d (p0v) d (p0w) \ 



dx dy dz J 



But the rate at which heat is going into the parallelipiped must be 

 equal to 



AxAyAzs d{pd) 



dt 



so that we obtain the equation 



kV°-0 



d(p0u) d ( P 0v) d (p0w) \ _ d_(p0) 



+ 



+ 



dx dy dz J dt 



We have also the 'equation of continuity' 



d ± + d(P M ) + d iEl} + 8 Q") = o 

 dt dx dy dz 



which reduces the above equation to 



kV - d - S P{ U dx + V dy + W dz) = pS di 



